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Chapter 8: Problem 5

For what values of \(r\) does the sequence \(\left\\{r^{n}\right\\}\) converge? Diverge?

Short Answer

Expert verified
Answer: The sequence \(\left\{r^n\right\}\) converges for \(r = 0, -1 < r < 1\) and diverges for \(r < -1\) and \(r > 1\).

Step by step solution

01

Consider the cases for \(r\)

We can consider three main cases: Case 1: \(r = 0\) Case 2: \(r > 0\) Case 3: \(r < 0\)
02

Analyze convergence for \(r = 0\)

When \(r = 0, r^n = 0^n\). The sequence becomes \(\left\{0, 0, 0, \dots \right\}\). Since all the terms in the sequence are equal to 0, the sequence converges to 0. Hence, for \(r = 0\), the sequence converges.
03

Analyze convergence for \(r > 0\)

When \(r > 0, r^n\) is a geometric sequence with a positive common ratio. For \(0 < r < 1\), the terms \(\left\{r^n\right\}\) decrease as \(n\) increases, and approach 0. Therefore, the sequence converges, in this case, to 0. However, if \(r > 1\), the terms of the sequence \(\left\{r^n\right\}\) increase as \(n\) increases and grow without any bound. Thus, the sequence diverges for \(r > 1\).
04

Analyze convergence for \(r < 0\)

When \(r < 0\), the sequence becomes an alternating sequence. For example, if \(r = -2\), the sequence is \(\left\{-2, 4, -8, 16, \dots\right\}\). If \(-1 < r < 0\), the terms of the sequence \(\left\{r^n\right\}\) will be alternating but decreasing in magnitude and approach 0. Hence, in this case, the sequence converges to 0. On the other hand, if \(r < -1\), the terms of the sequence will still alternate, but their absolute values will increase without any bound. Therefore, the sequence will diverge for \(r < -1\).
05

Summaries the results

From the above analysis, we can conclude: 1. The sequence \(\left\{r^n\right\}\) converges for the following \(r\) values: \(r = 0, -1 < r < 1\). 2. The sequence \(\left\{r^n\right\}\) diverges for the following \(r\) values: \(r < -1\) and \(r > 1\).

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Most popular questions from this chapter

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(\left|R_{n}\right|<10^{-6}\) ). Functions defined as series Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^{k} x^{k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 3^{-k}$$

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=\sqrt{n} \text { and } b_{n}=2 \ln n, n \geq 3$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100\)

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.
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